# Questions tagged [triangulated-categories]

A triangulated category is an additive category equipped with the additional structure of an autoequivalence (called the translation functor) and a class of of triangles satisfying certain axioms.

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### Computing Ext groups in a functor stable $\infty$-category

Let $I$ be a small category and $\mathcal{D}=D^b_\infty(\mathbb{Z})$ the bounded derived $\infty$-category of abelian groups. Consider the $\infty$-category $\mathcal{C}:=\mathrm{Fun}(I,\mathcal{D})$. ...

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### Can morphisms of Mayer-Vietoris triangles be completed into a $3\times 3$ square?

Let $(X,\mathcal{O}_X)$ be a topological ringed space and $(U,V)$ be an open covering of $X$ (i.e. $U$ and $V$ are two open subsets of $X$ such that $U\cup V=X$). Let $\mathcal{F}$ and $\mathcal{F}'$ ...

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### Cone of a morphism of complexes that are concentrated in degree $0$ and $1$

Let $R$ be a ring and $f:A\to A'$ and $g:B\to B'$ be morphisms of $R$-modules. Let $h:C_{\bullet}\to C_{\bullet}'$ be a morphism of $R$-module complexes fitting in a morphism of distinguished ...

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### Smallness condition for augmented algebras

I'm not sure this question is research level question. Sorry in advance.
Hypothesis
$k$ is a commutative ring.
$A$ is an augmented $k$-algebra.
$A^e$ is defined as the $k$-algebra $A\otimes_{k}A^{op}$...

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### Who introduced the heart ($\mathcal{C}^\heartsuit$) notation in the context of $t$-structures on triangulated categories?

In the context of $t$-structures
([Wikipedia],
[nLab],
[Notes I],
[Notes II],
[HA, Definition 1.2.1.11)],
[BBD, Définition 1.3.1]),
one often writes $\mathcal{C}^\heartsuit$ for the heart of a ...

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### Homotopy equivalences and Mapping Cones

Edit: Version 2:
Suppose that $A,B,C$ are chain complexes and $f: A \rightarrow B$ is a chain map. Suppose that there is a homotopy equivalence
$$ \text{Cone}(f: A \rightarrow B) \simeq C.$$
The chain ...

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### “Universal” triangulated category

Let $\mathcal{C}$ be some category. One way to map this category into a triangulated category is to take the category of simplicial objects $s\mathcal{C}$ (which is an $\infty$-category), take its ...

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### Proof in the realm of $\infty$ categories

I have recently started learning the language of $\infty$-categories. My approach is more to their use, rather than for their own sake. For this reason, as I feel I reached a good understanding of the ...

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### Does localization at quasi-isomorphisms imply homotopy invariance?

Usually, the derived category of some abelian category $A$ (I'm happy already with $A$-mod) is defined first taking chain complexes up to homotopy, and then localize at quasi-isomorphisms.
My question ...

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### Admissibility of intersection of subcategories

Let $\mathscr{T}$ be a triangulated category, and $\mathscr{A}$ be a right admissible subcategory, which means that $i_{\mathscr{A}} : \mathscr{A} \rightarrow \mathscr{T}$ has a right adjoint $i_{\...

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### idea and intuition behind triangulated category [closed]

I have some trouble in understanding the significance of some axiom of triangulated category.
if someone could explain me each axiom with some intuition,and explain me the intuition behind the ...

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### Why are Serre functors always exact?

Let $k$ be a field and $\mathcal{T}$ be a $k$-linear triangulated category with finite dimensional spaces of morphisms. Bondal and Kapranov proved that every Serre functor on $\mathcal{T}$ is exact (...

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### When is the homotopy category of an accessible $\infty$-category accessible?

Let $\mathcal C$ be an accessible $\infty$-category, and let $ho(\mathcal C)$ be its homotopy category. I can think of two "trivial" reasons for $ho(\mathcal C)$ to be accessible:
$ho(\mathcal C) = \...

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### The Octahedral Axiom in group theory

$\require{AMScd}$Here are two results about groups:
(The third isomorphism theorem) Suppose that I have $A \triangleleft B \triangleleft C$ and $A \triangleleft C$. Then $C/B \cong (C/A)/(B/A)$. ...

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### Morphism in a Verdier quotient

Let $\mathcal{T}$ be a triangulated category and take $\mathcal{S}$ a triangulated subcategory. Consider the Verdier quotient $\mathcal{T} \left/ \mathcal{S} \right.$, morphisms in this category are ...

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### A question on the proof of $D^b(coh(X))\simeq D^b_{coh}(Qcoh(X))$

Proposition 3.5 of "Fourier-Mukai Transforms in Algebraic Geometry" by Huybrechts claims that the is an equivalence of categories
$$
D^b(coh(X))\overset{\sim}{\to} D^b_{coh}(Qcoh(X))
$$
where $D^b(coh(...

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### Do we have $D^b_{coh}(X)\simeq D^b(coh(X))$ for a compact complex manifold $X$?

Let $X$ be a compact complex manifold and $\mathcal{O}_X$ be the structure sheaf of holomorphic functions. We call a sheaf of $\mathcal{O}_X$-module $\mathcal{F}$ coherent if it satisfies the ...

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### Can homotopy colimits recover cohomology sheaves?

The question is basically the one outlined in the title. Let $\mathcal{T}$ be a triangulated category containing infinite direct sums (e.g. $D_{qc}(X)$ for some separated, finite type over a field $k$,...

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### When is the heart of a triangulated category Grothendieck?

Are there conditions which guarantee that the heart of a triangulated category is Grothendieck? Is the compatibility between the t-structure with filtered colimits enough?

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### A concrete example of the deficiency of triangulated categories?

There seems to be a general sentiment that triangulated categories are not the "correct" notion to use because mapping cones of morphisms are unique, but only up to non-unique isomorphism.
Does ...

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### Equivalences between $D^b(\mathcal{B})$ and $\mathcal{D}_{\mathcal{B}}$, the triangulated category generated by $\mathcal{B}$

In the paper "Finite dimensional algebras and highest weight categories" of Cline, Parshall and Scott is stated as follows:
Let $\mathcal{B}$ be an abelian subcategory of a triangulated category $\...

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### Replacing triangulated categories with something better

Gelfand and Manin in their 1988 book on homological algebra write that the non-functoriality of cones means that "something is going wrong in the axioms of a triangulated category. Unfortunately at ...

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### Example of a tensor triangulated category with two different monoidal t-structures?

What's an example of a tensor triangulated category / symmetric monoidal stable $\infty$-category with two different monoidal $t$-structures?
While I'm at it: is there an example of a tensor ...

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### Computing a cone in a $\otimes$-triangulated category

I have a $\otimes$-triangulated category $\mathcal T$ and two triangles in $\mathcal T$:
$$
x_0\to x_1\to c_x\to \Sigma x_0\ \ \ \text{and}\ \ \ y_0\to y_1\to c_y\to \Sigma y_0.
$$
Consider the ...

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### Kontsevich's derived noncommutative geometry and Rosenberg's noncommutative 'spaces'

It appears to me (though I may be wrong) that the common opinion is that the main difference between two is that Rosenberg's version of noncommutative algebraic geometry mainly concerns as ...

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### Are hearts of all $t$-structures on smashing triangulated categories closed with respect to coproducts (also)?

Let $T$ be a triangulated category closed with respect to (small) coproducts, and $t$ be (an arbitrary!) a $t$-structure on $T$. I have noted that the heart $\underline{Ht}$ of $t$ is closed with ...

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### Homotopy pullbacks and pushouts in stable model categories

There are lots of similar questions that have been answered on this topic (particularly Homotopy limit-colimit diagrams in stable model categories), but I have a specific question that I do not ...

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### Which set of compact objects generates the subcategory of a compactly generated stable model category?

I couldn't find any info on what set of compact objects generates the following subcategory:
Let $k$ be a field of positive characteristic and let $G$ be either a finite group or a finite group ...

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### Are there universal homological functors?

There is a bifunctor $H: Stab^{op} \times Ab \to Top$ where $H(C,A)$ is the space of homological functors $C \to A$. Is this bifunctor left or right representable?
That is, for each small abelian ...

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### Abelianization derivator

About ten-fifteen years ago, when the theory of abstract triangulated categories reached a culminating point (after the publication of Neeman's book http://hopf.math.purdue.edu/Neeman/triangulatedcats....

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### Serre functors for non-proper categories

One usually defines a Serre functor to be a functor on a $k$-linear category $\mathcal{C}$ which has finite dimensional $Hom$s over $k$. In that case, the standard definition is that a Serre functor $...

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### Filtered triangulated category examples

I am reading Beilison Ginsburg Schechtman's "Koszul duality". In the section 1.3, they introduced the notion filtered triangulated categories with only one example, considering an abelian category ...

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### Weak generators of the right-bounded derived category of a finite-dimensional algebra

The setup:
Let $A$ be a finite-dimensional $k$-algebra over some field $k$.
Let $\mathcal{B} = Hot^-(Proj \, A)$ denote the homotopy category of cochain complexes of (possibly infinitely generated) ...

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### Filtrations of spectra related to cellular ones and singular homology

I would like to study filtrations of spectra (i.e., objects of the "topological" stable homotopy category $SH$; a filtration of a spectrum $E$ is a sequence of compatible maps $E_{\le i}\to E$) whose ...

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### When is $\Omega^1$ an equivalence?

Let $C$ be an abelian category with enough projectives and $\underline{C}$ the stable category of $C$ that is obtained by factoring out projective modules.
When is the functor $\Omega^1 : \underline{...

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### Is there any significance to Bousfield localization in the non-derived context?

The term "Bousfield localization" of a category $C$ is used in roughly two different ways:
There is a general usage (as in model categories or triangulated categories), which $\infty$-categorically ...

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### Highest weight category and weight structures

In various branches of representation theory, there is a notion of highest weight category.
On the other hand there is a notion of weight structure on a triangulated category $C$ (introduced by ...

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### Generators of unbounded derived categories of (quasi-)coherent sheaves

An object $T$ in a triangulated category $\mathcal{D}$ is called a generator if $T^\perp=0$, which means that for any nonzero $X$ in $\mathcal{D}$, there are $i\in\mathbb{Z}$ and a nonzero morphism $T[...

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### Why do we need the negative sign in TR2 (the “turning triangle” axiom) in the definition of triangulated categories?

In the TR2 of the definition of triangulated categories, we add a negative sign to an arrow when we turn the triangles. What is the significance/motivation of that negative sign ($-u[1]$ as in the ...

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### Vanishing natural transformation exact triangle

This question is a follow-up to this question I asked some time ago. Let $X$ be a smooth projective variety of dimension $n$ over $\mathbb{C}$. Let $\omega \in H^{n}(X,K_X)$, $\omega \neq 0$. Let
$$A ...

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### Is the derived category of $A$-dg-modules as a dg-category coincide with the ordinary definition of derived category?

Let $A$ be a unital dg-algebra over a base field $k$. We consider the category of (unbounded) right $A$-dg-modules with morphisms closed degree $0$ maps. We denote this category by dg-mod-$A$. We ...

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### If the homotopy category is well-generated, must the $\infty$-category be presentable?

Suppose $\mathcal{C}$ is a stable $\infty$-category whose homotopy category is a well-generated triangulated category in the sense of Neeman's book. Must $\mathcal{C}$ be a presentable $\infty$-...

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### A question on Voevodsky´s categories

I want to try to understand the Voevodsky´s big triangulated categories of motives $DM$ and $DM^{eff}$. Unfortunately, I am being not able to find answers to the following, too vague, questions:
1.- ...

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### (Middling) good morphisms of triangles

Neeman in his article "Some new axioms for triangulated categories" calls a morphism of distinguished triangles
$$\require{AMScd}
\begin{CD}
X @>>> Y @>>> Z @>>> X [1] \\
@...

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### Comparing self-equivalences of a triangulated category and automorphisms of its Grothendieck group

There is a homomorphism from the group of (isomorphism classes of) self-equivalences of a triangulated category to the automorphism group of its Grothendieck group. Is this homomorphism surjective? If ...

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### Prime spectrum of the derived category of holonomic $\mathcal{D}$-modules?

Let $X$ be a smooth algebraic (/projective if it simplifies things considerably) variety over $\mathbb{C}$ and consider the derived category $\mathcal{C}=D_h^b(\mathcal{D}_X)$ of bounded complexes of $...

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### Triangulated categories generated by a collection of submodules

In the book "Cohomology Rings of Finite Groups", by Carlson- Townsley - Elizondo there is the following corollary
The category $\mathsf{stmod}_{\mathbb{k}G}$ is generated as a triangulated category ...

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### Generating $K^b(\mathrm{proj})$ as a triangulated category from a full subcategory

Let $K^b(\mathrm{proj}\, A)$ be the bounded homotopy category of chain complexes over $\mathrm{proj}\, A$. In Rickard's paper 'Derived categories and stable equivalence', he defines a tilting complex ...

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### Extending a natural transformation using a distinguished triangle

$\require{AMScd}$
Let $\mathcal{T}$ be a triangulated category,
and $\mathcal{S}$ a full subcategory of $\mathcal{T}$ (which is not triangulated).
Let $F, G: \mathcal{T} \to \mathcal{T}$ be two ...

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### Techniques for Showing Triviality of K_1 of a Higher Category

Suppose $\cal{C}$ is a small stable $\infty$-category. Then, we have its K-theory spectrum $K(\cal{C})$ that gives us K-theory groups $K_n(\cal{C})$ by taking stable homotopy groups. There are ...